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International Conference on Typed Lambda Calculi and Applications

TLCA 2011: Typed Lambda Calculi and Applications pp 61–75Cite as

Game Semantics and Uniqueness of Type Inhabitance in the Simply-Typed λ-Calculus

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6690))

Abstract

The problem of characterizing sequents for which there is a unique proof in intuitionistic logic was first raised by Mints [Min77], initially studied in [BS82] and later in [Aot99]. We address this problem through game semantics and give a new and concise proof of [Aot99]. We also fully characterize a family of λ-terms for Aoto’s theorem. The use of games also leads to a new characterization of principal typings for simply-typed λ-terms. These results show that game models can help proving strong structural properties in the simply-typed λ-calculus.

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References

  1. Abramsky, S., McCusker, G.: Game semantics. In: Schwichtenberg, H., Berger, U. (eds.) Computational Logic: Proceedings of the 1997 Marktoberdorf Summer School, pp. 1–56. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  2. Aoto, T.: Uniqueness of normal proofs in implicational intuitionistic logic. Journal of Logic, Language and Information 8, 217–242 (1999)

    Article  MATH  Google Scholar 

  3. Barendregt, H.: λ-calculus: its syntax and semantics. Elsevier Science Publishers Ltd., Amsterdam (1984)

    MATH  Google Scholar 

  4. Broda, S., Damas, L.: On long normal inhabitants of a type. J. Log. Comput. 15(3), 353–390 (2005)

    Article  MATH  Google Scholar 

  5. Belnap, N.: The two-property. Relevance Logic Newsletter, 173–180 (1976)

    Google Scholar 

  6. Blass, A.: A game semantics for linear logic. Annals of Pure and Applied Logic 56, 183–220 (1992)

    Article  MATH  Google Scholar 

  7. Babaev, A., Soloviev, S.: A coherence theorem for canonical morphism in cartesian closed categories. Journal of Soviet Mathematics 20, 2263–2279 (1982)

    Article  MATH  Google Scholar 

  8. de Groote, P.: Towards abstract categorial grammars. In: Proceedings of the Conference Association for Computational Linguistics, 39th Annual Meeting and 10th Conference of the European Chapter, pp. 148–155 (2001)

    Google Scholar 

  9. Damas, L., Milner, R.: Principal type-schemes for functional programs. In: POPL 1982: Proceedings of the 9th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pp. 207–212. ACM, New York (1982)

    Google Scholar 

  10. Di Gianantonio, P., Franco, G., Honsell, F.: Game semantics for untyped λβ η-calculus. In: Girard, J.-Y. (ed.) TLCA 1999. LNCS, vol. 1581, pp. 114–128. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  11. Hindley, R.J.: Basic Simple Type Theory. Cambridge Press University, Cambridge (1997)

    Book  MATH  Google Scholar 

  12. Hirokawa, S.: Balanced formulas, minimal formulas and their proofs. Technical report, Research Institute of Fundamental Information Science, Kyochu University (1991)

    Google Scholar 

  13. Hyland, J.M.E., Ong, C.-H.L.: On full abstraction for PCF. In: Information and Computation, vol. 163(2), ch. 2, pp. 285–408. Elsevier Science B.V., Amsterdam (2000)

    Google Scholar 

  14. Huet, G.: Résolution d’équations dans les langages d’ordre 1,2,.,ω. PhD thesis, Université Paris 7 (1976)

    Google Scholar 

  15. Hughes, D.J.D.: Hypergame Semantics: Full Completeness for System F. PhD thesis, Oxford University (2000)

    Google Scholar 

  16. Jay, C.B., Ghani, N.: The virtues of η-expansion. J. of Functional Programming 5(2), 135–154 (1995); Also appeared as tech. report ECS-LFCS-92-243

    Article  MATH  Google Scholar 

  17. Kanazawa, M.: Parsing and generation as Datalog queries. In: Proceedings of the 45th Annual Meeting of the Association for Computational Linguistics, Prague, pp. 176–183. Association for Computational Linguistics (2007)

    Google Scholar 

  18. Ker, A.D., Nickau, H., Ong, L.: Innocent game models of untyped λ-calculus. Theoretical Computer Science 272(1-2), 247–292 (2002)

    Article  MATH  Google Scholar 

  19. Lorenzen, P.: Ein dialogisches konstruktivitatskriterium. Infinitistic Methods, 193–200 (1959)

    Google Scholar 

  20. Lorenz, K.: Dialogspiele als semantische grundlage von logikkalkiilen. Arch. Math. Logik Grundlag 11, 32–55 (1968)

    Article  MATH  Google Scholar 

  21. Mints, G.E.: Closed categories and the theory of proofs. Journal of Mathematical Sciences 15, 45–62 (1977)

    MATH  Google Scholar 

  22. Muskens, R.: Lambda Grammars and the Syntax-Semantics Interface. In: van Rooy, R., Stokhof, M. (eds.) Proceedings of the Thirteenth Amsterdam Colloquium, Amsterdam, pp. 150–155 (2001)

    Google Scholar 

  23. Nickau, H.: Hereditarily sequential functionals. In: Nerode, A., Matiyasevich, Y. (eds.) LFCS 1994. LNCS, vol. 813, pp. 253–264. Springer, Heidelberg (1994)

    Chapter  Google Scholar 

  24. Salvati, S.: On the membership problem for non-linear abstract categorial grammars. Journal of Logic, Language and Information 19(2), 163–183 (2010)

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. LaBRI - INRIA Sud-Ouest, 351, Cours de la Libération, 33405, Talence Cedex, France

    Pierre Bourreau & Sylvain Salvati

Authors
  1. Pierre Bourreau
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  2. Sylvain Salvati
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Editors and Affiliations

  1. Computing Laboratory, Oxford University, Wolfson Building, Parks Road, OX1 3QD, Oxford, UK

    Luke Ong

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© 2011 Springer-Verlag Berlin Heidelberg

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Bourreau, P., Salvati, S. (2011). Game Semantics and Uniqueness of Type Inhabitance in the Simply-Typed λ-Calculus. In: Ong, L. (eds) Typed Lambda Calculi and Applications. TLCA 2011. Lecture Notes in Computer Science, vol 6690. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21691-6_8

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  • DOI: https://doi.org/10.1007/978-3-642-21691-6_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-21690-9

  • Online ISBN: 978-3-642-21691-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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